PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 94(108) (2013), 111–124
DOI: 10.2298/PIM1308111D
ON RIEMANN AND WEYL COMPATIBLE TENSORS
Ryszard Deszcz, Małgorzata Głogowska, Jan Jełowicki,
Miroslava Petrović-Torgašev, and Georges Zafindratafa
Dedicated to Professor Witold Roter on his eighty-first birthday
Abstract. We investigate semi-Riemannian manifolds satisfying some curvature conditions. Those conditions are strongly related to pseudosymmetry.
1. Introduction
Let ∇, R, S, S, κ and C be the Levi-Civita connection, the Riemann–Christoffel
curvature tensor, the Ricci tensor, the Ricci operator, the scalar curvature and the
Weyl conformal curvature tensor of an n-dimensional semi-Riemannian manifold
(M, g), respectively. For precise definitions of the symbols used, we refer to Section 2 of this paper and [27] and [29].
Let A be a symmetric (0, 2)-tensor and B a generalized curvature tensor on a
manifold (M, g), n > 3. According to [72, Definition 3.1] (cf. [73, Definition 7.1])
the tensor A is called B-compatible if we have on M
(1.1)
B(AX, Y, Z, W ) + B(AZ, Y, W, X) + B(AW, Y, X, Z) = 0,
A is the endomorphism of the Lie algebra Ξ(M ) of vector fields on M defined by
(1.2)
g(AX, Y ) = A(X, Y ),
2010 Mathematics Subject Classification: Primary 53B20, 53B25, 53B30, 53B50; Secondary
53C40, 53C50, 53C80.
Key words and phrases: quasi-Einstein manifold, warped product, hypersurface, Chen ideal
submanifold, Riemann compatible tensor, Tachibana tensor, pseudosymmetry type condition.
The first named author was supported by the Serbian Academy of Sciences and Arts (SANU),
for the participation in the conference of XVII Geometrical Seminar, September 3-8, 2012, Zlatibor, Serbia. He was also supported by a grant of the Technische Universität Berlin, Germany. The
second named author was supported partially by SANU for the participation in that conference.
The second and third named authors also were supported by a grant of the Wrocław University
of Environmental and Life Sciences, Poland. The fourth named author thanks the Ministry of
Science of the Republic of Serbia, grant 174012, for support, and the Center for Scientific Research
of SANU and the University of Kragujevac for their partial support of her research done for this
paper.
111
112 DESZCZ, GŁOGOWSKA, JEŁOWICKI, PETROVIĆ-TORGAŠEV, AND ZAFINDRATAFA
and X, Y, Z, W ∈ Ξ(M ). In particular, a symmetric (0, 2)-tensor A on M is said
to be Riemann compatible (R-compatible) [73, Definition 1.1], Weyl compatible
(C-compatible) [74, Definition 2.1], respectively, if
(1.3)
R(AX, Y, Z, W ) + R(AZ, Y, W, X) + R(AW, Y, X, Z) = 0,
(1.4)
C(AX, Y, Z, W ) + C(AZ, Y, W, X) + C(AW, Y, X, Z) = 0,
holds on M , respectively. In [70, Theorem 3.5] (cf. [71, Theorem 4.14]) it was
proved that the Ricci tensor S of every Ricci-pseudosymmetric semi-Riemannian
manifold (R · S = LS Q(g, S), see Section 3) is R-compatible, i.e., we have on M
(1.5)
R(SX, Y, Z, W ) + R(SZ, Y, W, X) + R(SW, Y, X, Z) = 0.
This result was obtained already in [2, Lemma 3.3] and [28, Proposition 3.1(iv)]
(cf. [40, Lemma 2.4]). Unfortunately, [2], [28] and [40] are not cited in [70] and
[71]. We note that (1.5) was also obtained during the study on manifolds satisfying
some other curvature conditions of pseudosymmetry type: [8, Lemma 3.1, eq. (19)],
[12, Lemma 3.1, eq. (13); Proposition 3.1, eq. (22)], [38, Theorem 4.1, eq. (26)] and
[41, Proposition 3.9, eq. (43)]. If the Ricci tensor S of a semi-Riemannian manifold
(M, g), n > 4, is R-compatible, then also it is C-compatible [72, Proposition 3.4].
The converse statement is also true [74, Theorem 2.4].
In Section 3 we present definitions of quasi-Einstein, pseudosymmetric and
Ricci-pseudosymmetric manifolds. In particular, we present curvature properties
of manifolds with parallel Weyl tensor. In Section 4 we show that (1.1), and
in particular (1.3) and (1.5), are satisfied on certain semi-Riemannian manifolds
(Proposition 4.1, Theorems 4.1–4.4). Finally, in the last section we prove that
some warped products manifolds also satisfy (1.5) (Theorem 5.1, Remark 5.1).
2. Preliminaries
Throughout this paper, all manifolds (M, g) are assumed to be connected,
paracompact, manifolds of class C ∞ with the metric g of signature (s, n − s), 0 6
s 6 n. The manifold (M, g) will be called a semi(pseudo)-Riemannian manifold.
Clearly, if s = 0 or s = n then (M, g) is a Riemannian manifold. If s = 1 or s = n−1,
then (M, g) is a Lorentzian manifold. We define on M the endomorphisms X ∧A Y
and R(X, Y ) of the Lie algebra Ξ(M ) by (X ∧A Y )Z = A(Y, Z)X − A(X, Z)Y and
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z, respectively, where A is a symmetric
(0, 2)-tensor on M and X, Y, Z ∈ Ξ(M ). The Ricci tensor S, the Ricci operator S,
the scalar curvature κ and the endomorphism C(X, Y ) are defined by S(X, Y ) =
tr{Z 7→ R(Z, X)Y }, g(SX, Y ) = S(X, Y ), κ = tr S and
1 κ
X ∧g SY + SX ∧g Y −
X ∧g Y Z,
C(X, Y )Z = R(X, Y )Z −
n−2
n−1
respectively. The (0, 4)-tensors: G, R and C are defined by G(X1 , . . . , X4 ) =
g((X1 ∧g X2 )X3 , X4 ), R(X1 , . . . , X4 ) = g(R(X1 , X2 )X3 , X4 ), C(X1 , . . . , X4 ) =
g(C(X1 , X2 )X3 , X4 ), respectively, where X1 , X2 , . . . ∈ Ξ(M ). Further, we set
UR = {x ∈ M | R−(κ/((n−1)n))G 6= 0 at x}, US = {x ∈ M | S−(κ/n)g 6= 0 at x}
and UC = {x ∈ M | C 6= 0 at x}. We note that US ∪ UC = UR .
ON RIEMANN AND WEYL COMPATIBLE TENSORS
113
Let B(X1 , X2 ) be a skew-symmetric endomorphism of Ξ(M ) and B a (0, 4)tensor associated with B(X1 , X2 ) by
(2.1)
B(X1 , X2 , X3 , X4 ) = g(B(X1 , X2 )X3 , X4 ).
The tensor B is said to be a generalized curvature tensor if the following two
conditions are fulfilled: B(X1 , X2 , X3 , X4 ) = B(X3 , X4 , X1 , X2 ) and
B(X1 , X2 , X3 , X4 ) + B(X3 , X1 , X2 , X4 ) + B(X2 , X3 , X1 , X4 ) = 0.
For the symmetric (0, 2)-tensors E and F we define their Kulkarni–Nomizu product
E ∧ F (see, e.g., [25])
(E ∧ F )(X1 , X2 , X3 , X4 ) = E(X1 , X4 )F (X2 , X3 ) + E(X2 , X3 )F (X1 , X4 )
− E(X1 , X3 )F (X2 , X4 ) − E(X2 , X4 )F (X1 , X3 ).
The following tensors are generalized curvature tensors: R, C and E ∧ F , where E
and F are symmetric (0, 2)-tensors. We have G = 12 g ∧ g and
1
κ
(2.2)
C =R−
g∧S+
G.
n−2
(n − 2)(n − 1)
Let {e1 , e2 , . . . , en } be an orthonormal basis of Tx M at a point x ∈ M of a semiRiemannian manifold (M, g), n > 3, and let g(ej , ek ) = εj δjk , εj = ±1, and j, k ∈
{1, 2, . . . , n}. For a generalized curvature tensor B on M we denote by Ric(B), κ(B)
and Weyl(B) its scalar curvature, the Ricci tensor and
Pnthe Weyl tensor, respectively.
Thus at every x ∈ M we have: Ric(B)(X, Y ) = j=1 εj B(ej , X, Y, ej ), κ(B) =
Pn
j=1 εj Ric(B)(ej , ej ) and
(2.3)
Weyl(B) = B −
1
κ(B)
g ∧ Ric(B) +
G.
n−2
(n − 2)(n − 1)
Lemma 2.1. [22, Lemma 2(ii)]; cf. [50, p. 48]; The Weyl tensor Weyl(B) of
any generalized curvature tensor B on a 3-dimensional semi-Riemannian manifold
(M, g) vanishes, i.e., on M we have B = g ∧ Ric(B) − (κ(B)/2) G.
Let B(X, Y ) be a skew-symmetric endomorphism of Ξ(M ), and let B be the
tensor defined by (2.1). We extend the endomorphism B(X, Y ) to a derivation
B(X, Y )· of the algebra of tensor fields on M , assuming that it commutes with
contractions and B(X, Y ) · f = 0 for any smooth function f on M . Now for a
(0, k)-tensor field T , k > 1, we can define the (0, k + 2)-tensor B · T by
(B · T )(X1 , . . . , Xk , X, Y ) = (B(X, Y ) · T )(X1 , . . . , Xk )
= −T (B(X, Y )X1 , X2 , . . . , Xk ) − · · · − T (X1 , . . . , Xk−1 , B(X, Y )Xk ).
If A is a symmetric (0, 2)-tensor, then we define the (0, k + 2)-tensor Q(A, T ) by
Q(A, T )(X1 , . . . , Xk , X, Y ) = (X ∧A Y · T )(X1 , . . . , Xk )
= −T ((X ∧A Y )X1 , X2 , . . . , Xk ) − · · · − T (X1 , . . . , Xk−1 , (X ∧A Y )Xk ).
In this manner we obtain the (0, 6)-tensors B · B and Q(A, B). Substituting B = R
or B = C, T = R or T = C or T = S, A = g or A = S in the above formulas, we
get the tensors R · R, R · C, C · R, R · S, Q(g, R), Q(S, R), Q(g, C) and Q(g, S).
114 DESZCZ, GŁOGOWSKA, JEŁOWICKI, PETROVIĆ-TORGAŠEV, AND ZAFINDRATAFA
Let A be a symmetric (0, 2)-tensor and T a (0, k)-tensor, k > 2. Following [32],
we will call the tensor Q(A, T ) the Tachibana tensor of A and T , or the Tachibana
tensor for short. We would like to point out that in some papers, the tensor Q(g, R)
is called the Tachibana tensor (see, e.g., [57, 61, 62, 81]).
Let Bhijk , Thijk , Aij , (B · T )hijklm and Q(A, T )hijklm , h, i, . . . , m ∈ {1, . . . , n},
be the local components of the generalized curvature tensors B and T , a symmetric
(0, 2)-tensor A and the tensors B · T and Q(A, T ), respectively. We have [32]
(B · T )hijklm = g rs (Trijk Bshlm + Thrjk Bsilm
+ Thirk Bsjlm + Thijr Bsklm ),
(2.4)
g rs (B · T )hrsklm = g rs (Ric(T )kr Bshlm + Ric(T )hr Bsklm ),
Q(A, T )hijklm = Ahl Tmijk + Ail Thmjk + Ajl Thimk + Akl Thijm
(2.5)
− Ahm Tlijk − Aim Thljk − Ajm Thilk − Akm Thijl ,
g Q(A, T )hrsklm = Asl Tskhm − Asl Tshmk − Asm Tskhl + Asm Tshlk
rs
+ Q(A, Ric(T ))hklm .
Let A be a symmetric (0, 2)-tensor on a semi-Riemannian manifold (M, g),
n > 3. We define the tensors A0 , A1 , Ap , p > 2, and the endomorphisms (cf.,
[82, 83]) A0 , A1 , Ap , p > 2, by A0 = g, A1 = A, Ap (X, Y ) = Ap−1 (AX, Y ) and
A0 = Id, A1 = A, Ap X = Ap−1 (AX), respectively, where A is the endomorphism
related to A by (1.2) and Id the identity transformation of Ξ(M ).
Using the above presented definitions we can prove the following
Proposition 2.1. If A is a symmetric (0, 2)-tensor and B a generalized curvature tensor on a semi-Riemannian manifold (M, g), n > 3, expressed by a linear
combination of the tensors Ap1 ∧ Ap2 , p1 , p2 > 0, then Ap , p > 0, are B-compatible.
Let H be the second fundamental tensor of a hypersurface M , dim M > 3,
isometrically immersed in a conformally flat semi-Riemannian manifold N . Using
Proposition 2.1 and identity (20) of [47] (cf. [37, Section 4]) we can easily prove
that the tensors H p , p > 0, are Weyl compatible.
Semi-Riemannian manifolds (M, g), n > 4, admitting generalized curvature
tensors expressed by a linear combination of the tensors: A∧A, g∧A and g∧g, where
A is a symmetric (0, 2)-tensor on M , were investigated in [65]. In particular, [65]
contains results on non-quasi Einstein and non-conformally flat manifolds having
the Riemann–Christoffel curvature tensor expressed by a linear combination of the
tensors S ∧ S, g ∧ S and g ∧ g. Semi-Riemannian manifolds with this property are
called Roter type manifolds, see [27] and [53] and references therein.
Example 2.1. We define on M = {(x, y, z, t) : x > 0, y > 0, z > 0, t > 0} ⊂ R4
the metric tensor g by ds2 = exp(y) dx2 + (x z)2 dy 2 + dz 2 − dt2 . The Ricci tensor
S of (M, g) is expressed by a linear combination of g and some other symmetric
(0, 2)-tensors [9, Section 4]. Since g is a product metric of some 3-dimensional
and an 1-dimensional metric, the equality R · R = Q(S, R) is satisfied on M [11,
Corollary 3.2]. We also have on M : κ = 1/(2 x2 z 2 ), rank(S) = · · · = rank(S 4 ) = 3,
ON RIEMANN AND WEYL COMPATIBLE TENSORS
115
and
Q(S, S 2 ∧ S 2 ) = Q(S 3 − exp(y)/(2xz 2 )S 2 , S ∧ S),
R = φ1 S ∧ S + φ2 S ∧ S 2 + φ3 S 2 ∧ S 2 ,
ω(X)R(Y, Z) + ω(Y )R(Z, X) + ω(Z)R(X, Y ) = 0,
φ1 = (16x2 z 4 + z 2 (4x2 + 1) exp(y))/(8z 2 + 2 exp(y)),
φ2 = −4x2 z 4 exp(y)/(4z 2 + exp(y)), φ3 = 8x4 z 6 exp(y)/(4z 2 + exp(y)),
where the 1-form ω is defined by ω(∂x ) = ω(∂y ) = 1, ω(∂z ) = ω(∂t ) = 0. Finally,
from Proposition 2.1 it follows that the tensors S p , p > 0, are R-compatible.
3. Some special classes of semi-Riemannian manifolds
A semi-Riemannian manifold (M, g), n > 2, is said to be an Einstein manifold
if its Ricci tensor S is proportional to g, i.e., on M we have S = nκ g, where κ is
the scalar curvature. It is well-known that the scalar curvature κ of an Einstein
manifold of dimension > 3 is a constant. A semi-Riemannian manifold (M, g),
n > 3, is called a quasi-Einstein manifold if at every x ∈ M its Ricci tensor satisfies
rank(S − αg) 6 1, for some α ∈ R, i.e., the condition S = αg + εw ⊗ w, for some
α ∈ R, ε = ±1, w ∈ Tx∗ M holds at every x ∈ US ⊂ M (see, e.g., [39, 43, 54]).
Evidently, w is non-zero at every point of US . It is well-known that quasi-Einstein
manifolds arose during the study of exact solutions of the Einstein field equations
as well as during considerations of quasi-umbilical hypersurfaces of conformally flat
spaces. We refer to [24, 27, 28, 37, 41, 42, 43, 54] for results on quasi-Einstein
hypersurfaces in spaces of constant curvature. Recently, quasi-Einstein manifolds
were investigated amongst others in [49, 63, 64, 68].
An extension of the class of Einstein manifolds form Ricci-symmetric manifolds,
i.e., manifolds of dimension > 3 with ∇S = 0. An important subclass of the class of
Ricci-symmetric manifolds form locally symmetric manifolds, i.e., manifolds with
∇R = 0. The last two equations lead to the integrability conditions
(3.1)
(a) R · S = 0,
(b) R · R = 0,
respectively. Semi-Riemannian manifolds satisfying (3.1)(a) and (3.1)(b) are called
Ricci-semisymmetric and semisymmetric [84], respectively. Any semisymmetric
manifold is Ricci-semisymmetric. It is known that the converse statement is not
true. Semisymmetric Riemannian manifolds were classified in [84]. Ricci-semisymmetric Riemannian manifolds were investigated, amongst others, in [79], see also
[69, 80]. In those papers Ricci-semisymmetric manifolds (submanifolds) are called
Ric-semisymmetric manifolds (submanifolds).
We consider now non-Riemannian semi-Riemannian manifolds (M, g), n > 4,
with parallel Weyl tensor (∇C = 0), which are in addition non-locally symmetric
(∇R 6= 0) and non-conformally flat (C 6= 0). Such manifolds are called essentially
conformally symmetric manifolds, e.c.s. manifolds, in short (see e.g., [15, 16]).
E.c.s. manifolds are semisymmetric manifolds satisfying κ = 0 and Q(S, C) = 0
[15, Theorems 7, 8 and 9]. In addition, on every e.c.s. manifold (M, g) we have [16]
116 DESZCZ, GŁOGOWSKA, JEŁOWICKI, PETROVIĆ-TORGAŠEV, AND ZAFINDRATAFA
rank S 6 2 and F C = 12 S ∧ S, where F is a function on M , called the fundamental
function. Also the local structure of e.c.s. manifolds is determined [17, 19]. Certain
e.c.s. metrics are realized on compact manifolds [18, 20]. E.c.s. warped products
were investigated in [59].
A semi-Riemannian manifold (M, g), n > 3, is said to be pseudosymmetric [33]
if the tensors R · R and Q(g, R) are linearly dependent at every point of M . This
is equivalent on UR ⊂ M to
(3.2)
R · R = LR Q(g, R),
where LR is a function on this set. A pseudosymmetric manifold is called a pseudosymmetric space of constant type if the function LR is constant [4, 66]. We mention
that [33] is the first publication, in which a semi-Riemannian manifold satisfying
(3.2) was called the pseudosymmetric manifold. However results on manifolds satisfying (3.2) also are contained in some papers published earlier than [33] (see, e.g.,
[1, 55, 78]). For instance, in [55, proof of Lemma 3] it was stated that fibres
of semisymmetric warped products satisfy (3.2). We note that (3.2) is equivalent
to (R − LR G) · (R − LR G) = 0. Such expression of (3.2) was used in [78]. Evidently, any semisymmetric manifold is pseudosymmetric. The converse statement
is not true. For instance, the Schwarzschild spacetime, the Kottler spacetime and
the Reissner-Nordström spacetime satisfy (3.2) with non-zero function LR [48] (see
also [34, 56]). It is well-known that the Schwarzschild spacetime was discovered
in 1916 by Schwarzschild, during his study on solutions of Einstein’s equations.
It seems that the Schwarzschild spacetime, the Reissner–Nordström spacetime,
as well as some Friedmann–Lemaître–Robertson–Walker spacetimes are the “oldest” examples of a non-semisymmetric pseudosymmetric warped product manifolds
(cf. [35]). We also mention that Roter type manifolds are non-quasi-Einstein and
non-conformally flat pseudosymmetric (see, e.g., [27, 53]).
A semi-Riemannian manifold (M, g), n > 3, is said to be Ricci-pseudosymmetric
[21, 36] if the tensors R · S and Q(g, S) are linearly dependent at every point of
M . This is equivalent on US ⊂ M to
(3.3)
R · S = LS Q(g, S),
where LS is some function on this set. A Ricci-pseudosymmetric manifold is called
a Ricci-pseudosymmetric manifold of constant type if the function LS is constant
[52]. We note that (3.2) implies (3.3). The converse statement is not true, provided that n > 4, (see, e.g., [27, 32]). However, 3.2 and 3.3 are equivalent on every
3-dimensional manifold. Ricci-pseudosymmetric warped product manifolds were investigated, amongst others, in [7, 21, 36, 46]. An example of quasi-Einstein pseudosymmetric, resp. non-pseudosymmetric Ricci-pseudosymmetric, warped product
manifold are given in [37], respectively [43]. Recently in [60] Ricci-semisymmetric
and Ricci-pseudosymmetric Riemannian manifolds were called Riemannian manifolds having semi-parallel Ricci operator S, R(X, Y ) · S = 0, and pseudo-parallel
Ricci operator S, R(X, Y ) · S = L (X ∧ Y ) · S, respectively, where L is a function on
M and X, Y ∈ Ξ(M ). Evidently, the last two conditions are equivalent to (3.1)(a)
and (3.3), respectively.
ON RIEMANN AND WEYL COMPATIBLE TENSORS
117
We refer to [23, 28, 35, 57, 58, 61, 62] for further results related to those
classes of manifolds. We mention only that a geometrical interpretation of (3.2)
and (3.3), in the Riemannian case, is given in [57] and [62], respectively.
4. Riemann compatible tensors
Lemma 4.1. Let A be a symmetric (0, 2)-tensor and B, T and T1 generalized
curvature tensors on a semi-Riemannian manifold (M, g), n > 4, satisfying on M
the condition B · T = Q(A, T ) + L Q(g, T1 ), where L is a function. Then
(4.1) B(T X, Y, Z, W ) + B(T Z, Y, W, X) + B(T W, Y, X, Z)
+ 3(T (AX, Y, Z, W ) + T (AZ, Y, W, X) + T (AW, Y, X, Z)) = 0
holds on M , where A is defined by (1.2) and T by g(T X, Y ) = Ric(T )(X, Y ).
Proof. From the equation (B · T )hijklm = Q(A, T )hijklm + LQ(g, T1 )hijklm ,
by contraction with g ij and making use of (2.4) and (2.5), we get
(4.2) Ths Bsklm + Tks Bshlm = Q(A, Ric(T ))hklm + L Q(g, Ric(T1 ))hklm
− Asl Rskmh − Asm Rskhl − Asl Rshmk − Asm Rshkl ,
Summing (4.2) cyclically in h, l, m we obtain
Ths Bsklm + Tls Bskmh + Tms Bskhl + 2(Ash Tsklm + Asl Tskmh + Asm Tskhl )
= Ash (Tsmkl + Tslmk ) + Asl (Tshkm + Tsmhk ) + Asm (Tslkh + Tshlk ),
Ths Bsklm + Tls Bskmh + Tms Bskhl + 3(Ash Tsklm + Asl Tskmh + Asm Tskhl ) = 0,
completing the proof.
Similarly, we also can prove the following
Lemma 4.2. If A, A1 and A2 are symmetric (0, 2)-tensors and B a generalized
curvature tensor on a semi-Riemannian manifold (M, g), n > 4, satisfying on M
the condition B · A = Q(A1 , A2 ), then (1.1) holds on M .
As an immediate consequence of Lemma 2.1 we have
Lemma 4.3. If A is a symmetric (0, 2)-tensor and T a generalized curvature
tensor on a semi-Riemannian manifold (M, g), n = 3, then we have on M
(4.3) T (AX, Y, Z, W ) + T (AZ, Y, W, X) + T (AW, Y, X, Z)
= g(X, Y )D(W, Z) + g(Z, Y )D(X, W ) + g(W, Y )D(Z, X),
T (T X, Y, Z, W ) + T (T Z, Y, W, X) + T (T W, Y, X, Z) = 0,
where D(X, Y ) = Ric(T )(AX, Y ) − Ric(T )(AY, X), T is defined in Lemma 4.1 and
A by (1.2).
From the above lemmas it follows
118 DESZCZ, GŁOGOWSKA, JEŁOWICKI, PETROVIĆ-TORGAŠEV, AND ZAFINDRATAFA
Proposition 4.1. Let (M, g), n > 4, be a semi-Riemannian manifold.
(i) If B and T are generalized curvature tensors on M satisfying on M
(4.4)
B · T = Q(Ric(T ), T ) + LQ(g, Weyl(T )),
where L is a function, then
(4.5) B(T X, Y, Z, W ) + B(T Z, Y, W, X) + B(T W, Y, X, Z)
+ 3(T (T X, Y, Z, W ) + T (T Z, Y, W, X) + T (T W, Y, X, Z)) = 0
holds on M , where T is defined in Lemma 4.1.
(ii) [32, Proposition 2.1] If the following condition is satisfied on M
(4.6)
R · R = Q(S, R) + LQ(g, C),
where L is a function, then (1.5) holds on M .
(iii) [10, Lemma 2.2(i)] If the following condition is satisfied on M
(4.7)
R · R = LQ(S, R),
where L is a function, then we have on M
(1 + 3L)(R(SX, Y, Z, W ) + R(SZ, Y, W, X) + R(SW, Y, X, Z)) = 0.
(iv) [32, Remark 2.1] (1.5) is satisfied on any 3-dimensional manifold (M, g).
As it was shown in [77, Theorems 2.2 and 2.5], some curvature 2-forms on a
Riemannian manifold (M, g) are closed if and only if (1.5) holds on M . For further
results related to the questions related to the closedness of some forms and (1.5) see
[71, Theorem 4.2], [75, Theorem 6.2] or [76, Theorem 3.4]. We mention that the
result presented in Proposition 4.1(iii), i.e., Lemma 2.2(i) of [10], was also proved
in [71, Theorem 4.17]. However, Lemma 2.2(i) of [10] is not cited in [71]. Similarly,
the result presented in Proposition 4.1(iv), i.e., Remark 2.1 of [32], was also proved
in [73, ] (see Section 5.1). Unfortunately, [32] is not cited in [73].
Let (M, g), n > 3, be a semi-Riemannian manifold satisfying the condition
(4.8)
R · R = LQ(S p , R),
p > 0,
where L is a function on M . From Lemma 4.1 it follows that (4.8) implies (1.3),
with A = S + LS p . We mention that special para-Sasakian Riemannian manifolds
satisfying (4.8) were investigated in [82, 83]. For instance, in [82] it was proved
that such manifolds, under some additional assumptions, are the spaces of quasi
constant curvature. Thus, in particular, they are quasi-Einstein manifolds.
Let M be a hypersurface isometrically immersed in a a semi-Riemannian space
of constant curvature Nsn+1 (c), with signature (s, n + 1 − s), n > 4, where c =
κ
e/(n(n + 1)) and κ
e are the sectional and the scalar curvature of the ambient space,
respectively. It is known that R · R = Q(S, R) − ((n − 2)e
κ)/(n(n + 1))Q(g, C) holds
on M [47]. Now Proposition 4.1(ii) implies (cf. [41, eq. (43)])
Theorem 4.1. (1.5) holds on every hypersurface M in Nsn+1 (c), n > 4.
Chen ideal submanifolds M isometrically immersed in Euclidean spaces [5, 6],
satisfying some conditions of pseudosymmetry type, were investigated in [31, 44,
45]. Using equations (26.1)–(26.4) of [31] we can easily prove the following
ON RIEMANN AND WEYL COMPATIBLE TENSORS
119
Theorem 4.2. (1.5) holds on every Chen ideal submanifold M , of dimension
> 4, isometrically immersed in a Euclidean space.
For a (0, 6)-tensor T on M we denote by
X
T (X1 , X2 , X3 , X4 , X5 , X6 )
(X1 ,X2 ),(X3 ,X4 ),(X5 ,X6 )
the sum T (X1 , X2 , . . . , X6 ) + T (X3 , X4 , . . . , X6 , X1 , X2 ) + T (X5 , X6 , X1 , . . . , X4 ),
where X1 , . . . , X4 , X, Y ∈ Ξ(M ). It is well-known that on every semi-Riemannian
manifold (M, g) the following identity, called the Walker identity, is satisfied
X
(4.9)
(R · R)(X1 , X2 , X3 , X4 , X, Y ) = 0.
(X1 ,X2 ),(X3 ,X4 ),(X,Y )
We can also investigate semi-Riemannian manifolds, of dimension > 4, satisfying:
X
(4.10)
(R · C)(X1 , X2 , X3 , X4 , X, Y ) = 0,
(X1 ,X2 ),(X3 ,X4 ),(X,Y )
X
(4.11)
(X1 ,X2 ),(X3 ,X4 ),(X,Y )
(4.12)
X
(X1 ,X2 ),(X3 ,X4 ),(X,Y )
(C · R)(X1 , X2 , X3 , X4 , X, Y ) = 0,
(R · C − C · R)(X1 , X2 , X3 , X4 , X, Y ) = 0.
We mention that hypersurfaces in spaces of constant curvature satisfying (4.10)–
(4.12) were investigated in [29], [43] and [51]. We also have
Proposition 4.2. Let (M, g), n > 4, be a semi-Riemannian manifold.
(i) [14, Lemma 1] For
Pa symmetric (0, 2)-tensor A and a generalized curvature
tensor B on M we have (X1 ,X2 ),(X3 ,X4 ),(X,Y ) Q(A, B)(X1 , X2 , X3 , X4 , X, Y ) = 0.
(ii) [29, Proposition 4.1] The conditions (4.10)–(4.12) are equivalent.
It is easy to check that every pseudosymmetric manifold satisfies (4.10). More
generally, in [7, Theorem 2.3] it was proved that (4.10) holds on any Ricci-pseudosymmetric manifold. In that paper it was proved that R ·S = −(1/n)Q(g, A), holds
on any semi-Riemannian manifold, of dimension > 5, satisfying (4.10), where A is
the (0, 2)-tensor with the local components Aij = g hk (R · S)hijk . Thus we have
Theorem 4.3. (1.5) holds on every manifold (M, g), n > 5, satisfying (4.10).
In the next section we also prove that (1.5) holds on any 4-dimensional warped
product satisfying (4.10). In addition, Proposition 4.2 and Theorem 4.3 imply
Theorem 4.4. Let (M, g), n > 5, be a semi-Riemannian manifold. If the
tensor R · C, or C · R, or R · C − C · R, is expressed on M by a linear combination
of the Tachibana tensors of the form Q(A, B), where A is a symmetric (0, 2)-tensor
and B a generalized curvature tensor, then (1.5) holds on M .
120 DESZCZ, GŁOGOWSKA, JEŁOWICKI, PETROVIĆ-TORGAŠEV, AND ZAFINDRATAFA
5. Warped products with Riemann compatible Ricci tensor
e , ge), dim M = p, dim N = n − p, 1 6 p < n, be semiLet now (M , g) and (N
Riemannian manifolds covered by systems of charts {U ; xa } and {V ; y α }, respece
tively. Let F be a positive smooth function on M . The warped product M ×F N
e
e
of (M , g) and (N , e
g) is the product manifold M × N with the metric g = g ×F ge
e −→ M
g, where π1 : M × N
defined by ([3], [67]) g ×F ge = π1∗ g + (F ◦ π1 ) π2∗ e
e −→ N
e are the natural projections on M and N
e , respectively.
and π2 : M × N
e.
Let {U × V ; x1 , . . . , xp , xp+1 = y 1 , . . . , xn = y n−p } be a product chart for M × N
The local components gij of the metric g = g ×F e
g with respect to this chart are
the following gij = gab if i = a and j = b, gij = F e
gαβ if i = α and j = β, and
gij = 0 otherwise, where a, b, c, d, e, f ∈ {1, . . . , p}, α, β, γ, δ, ε, µ ∈ {p + 1, . . . , n}
and h, i, j, k, l, m, r, s ∈ {1, 2, . . . , n}. We will denote by bars (resp., by tildes)
g).
tensors formed from g (resp., e
The local components of the Riemann–Christoffel curvature tensor R and the
e which
local components Sij of the Ricci tensor S of the warped product M ×F N
may not vanish identically are the following (see, e.g., [22, 25, 48])
(5.1)
(5.2)
1
eαβγβ − ∆1 F G
e αβγδ ,
Rabcd = Rabcd , Rαabβ = − Tab geαβ , Rαβγδ = F R
2
4
1
n−p
n−p−1
Tab , Sαβ = Seαβ −
gab Tab +
∆1 F geαβ ,
Sab = S ab −
2F
2
4F
1
where ∆1 F = ∆1g F = g ab Fa Fb , Tab = ∇b Fa − 2F
Fa Fb and Fa = (∂F )/(∂xa ). Fur√
√
√
√
ther, let Hess( F ) be the Hessian of F . We have (Hess( F ))ab = 1/(2 F ) Tab .
Using now (5.1) and (5.2) we can easily prove the following
e satisfies (1.5) if and only if
Proposition 5.1. The manifold M ×F N
(5.3)
(5.4)
(5.5)
g ef (S ae Rf bcd + S ce Rf bda + S de Rf bac )
n − p ef
−
g (T ae Rf bcd + T ce Rf bda + T de Rf bac ) = 0,
2F
√
√
g ef (S de (Hess( F ))f a − S ae (Hess( F ))f d ) = 0,
eµβγδ + Seγε R
eµβδα + Seδε R
eµβαγ ) = 0.
geεµ (Seαε R
As an immediate consequence of propositions 4.1 (iv) and 5.1 we get
e , dim M = p 6 2, dim N
e = n − p 6 3,
Theorem 5.1. (i) The manifold M ×F N
satisfies (1.5).
e , ge) a manifold
(ii) If (M , g) is an p-dimensional manifold, p 6 2, and (N
e
satisfying (5.5), then (1.5) holds on M ×F N .
e, e
(iii) If (M , g), dim M = 3, and (N
g) are manifolds satisfying (5.4) and (5.5),
e.
respectively, then (1.5) holds on M ×F N
e , ge) a
(iv) If (M , g), dim M = p > 3, is a space of constant curvature and (N
e
manifold satisfying (5.5), then (1.5) holds on M ×F N .
ON RIEMANN AND WEYL COMPATIBLE TENSORS
121
Remark 5.1. (i) From Theorem 5.1(i) it follows that any 4-dimensional warped
product, with 1-dimensional base manifold (M , g), satisfies (1.5). In addition such
warped product also satisfies (4.6) [13, Theorem 4.1]. Thus in particular, every
generalized Robertson-Walker spacetime satisfies (1.5) and (4.6).
(ii) From Theorem 5.1(ii) it follows that if (M , g) is an 2-dimensional manifold
e, e
and (N
g) an (n − p)-dimensional semi-Riemannian space of constant curvature,
e . Such warped product is a manifold
n − p > 2, then (1.5) holds on M ×F N
with pseudosymmetric Weyl tensor [30], i.e., the condition C · C = LC Q(g, C) is
satisfied, where LC is a function.
e g),
(iii) From Proposition 5.1 it follows that if (M , g),
√ dim M = p > 2, and (N , e
e
dim N = n − p > 2, are Einstein manifolds and Hess( F ) is proportional to g, then
e.
(1.5) holds on M ×F N
(iv) In the previous section we proved that (1.5) holds on any manifold, of dimension > 5, satisfying (4.10). The condition (1.5) also holds on any 4-dimensional
warped product satisfying (4.10). This is a consequence of Theorem 5.1 (i) and (iii)
and the fact that (5.4) holds on any warped product satisfying (4.10) [7].
Example 5.1. We define on M = {(x, y, w, z) : x > 0, y > 0, w > 0, z > 0} ⊂
R4 the family of warped product metrics by
ds2 = xα1 y β1 wγ1 dx2 + xα2 y β2 wγ2 dy 2 + xα3 y β3 wγ3 dw2 + xα4 y β4 wγ4 dz 2 ,
where α1 , α2 , . . . , γ4 ∈ R. Certain metrics of that family do not satisfy (1.5).
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Dept. of Math., Wrocław University of Environmental and Life Sci., Wrocław, Poland
Ryszard.Deszcz@up.wroc.pl
Malgorzata.Glogowska@up.wroc.pl
Jan.Jelowicki@up.wroc.pl
Dept. of Math., Faculty of Science, University of Kragujevac, Kragujevac, Serbia
mirapt@kg.ac.rs
Laboratoire de Mathématiques et leurs Applications de Valenciennes
Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, Lamath-ISTV2
59313 Valenciennes cedex 9, France
Georges.Zafindratafa@univ-valenciennes.fr